285 is summation of 3 numbers. Ratio between 2nd and 3rd numbers is 6:5. Ratio between 1st and 2nd numbers is 3:7. The 3rd number is?

Options

A. a. 135
B. b. 150
C. c. 124
D. d. 105

Answer

To find the third number given the conditions in the problem, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the three numbers be \( A \), \( B \), and \( C \).

2. **Set Up the Ratios**:
- The ratio between the 2nd and 3rd numbers is given as \( \frac{B}{C} = \frac{6}{5} \).
- The ratio between the 1st and 2nd numbers is given as \( \frac{A}{B} = \frac{3}{7} \).

3. **Express B and C in Terms of a Common Variable**:
From the ratio \( \frac{B}{C} = \frac{6}{5} \), we can express:
\[ B = 6k \quad \text{and} \quad C = 5k \]
for some variable \( k \).

4. **Express A in Terms of B**:
From the ratio \( \frac{A}{B} = \frac{3}{7} \), we can express:
\[ A = \frac{3}{7}B = \frac{3}{7}(6k) = \frac{18k}{7} \]

5. **Set Up the Equation for the Sum**:
According to the problem, the sum of the three numbers is 285:
\[ A + B + C = 285 \]
Substituting the expressions for \( A \), \( B \), and \( C \):
\[ \frac{18k}{7} + 6k + 5k = 285 \]

6. **Combine the Terms**:
To combine the terms, we first convert \( 6k \) and \( 5k \) into fractions with a common denominator of 7:
\[ \frac{18k}{7} + \frac{42k}{7} + \frac{35k}{7} = 285 \]
This simplifies to:
\[ \frac{18k + 42k + 35k}{7} = 285 \]
\[ \frac{95k}{7} = 285 \]

7. **Solve for k**:
Multiply both sides by 7:
\[ 95k = 285 \times 7 \]
\[ 95k = 1995 \]
Now, divide by 95:
\[ k = \frac{1995}{95} = 21 \]

8. **Find the Values of A, B, and C**:
Now that we have \( k \), we can find \( A \), \( B \), and \( C \):
\[ B = 6k = 6 \times 21 = 126 \]
\[ C = 5k = 5 \times 21 = 105 \]
\[ A = \frac{18k}{7} = \frac{18 \times 21}{7} = 54 \]

9. **Conclusion**:
The third number \( C \) is:
\[ C = 105 \]

### Final Answer:
The third number is **105**.